Coaxial cable

For ideal coaxial conductors of radii $a<b$ filled with a homogeneous permittivity $ϵ$ and magnetic permeability $\mu$, Gauss's law gives $E_r=\lambda /(2\pi ϵr)$ and hence $C^{\prime }=2\pi ϵ/\ln (b/a)$. The magnetic field $H_{\phi }=I/(2\pi r)$ has energy per unit length $L^{\prime }{I}^2/2$, giving $L^{\prime }=\mu \ln (b/a)/(2\pi )$. These are the external-field parameters when internal magnetic penetration into the conductors is neglected. Their product is $L^{\prime }{C}^{\prime }=\mu ϵ$, and the characteristic impedance is $\sqrt{L^{\prime }/C^{\prime }}=\sqrt{\mu /ϵ}\ln (b/a)/(2\pi )$.